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Full text of "Analytical Mechanics"

WORK                                       179
It will be observed that the torsional rigidity of a solid shaft varies directly as the fourth power of the radius and inversely as the length.
148. Work Done in Twisting a Rod.  Work done by a torque is obtained by substituting the expression for the torque in the work equation. Thus
r = I
JQ
Gde.
Q
[byeq. (9).] (11)
where
PROBLEMS.
1.   What are the dimensions of stress, strain, and modulus of elasticity?
2.   A steel rod of J-inch radius is found to stretch 0.004 inch in 10 inches of its length when a load of 10,000 pounds is gradually applied. Find the Young's modulus of the rod.
3.   The Young's modulus of a brass wire is 10.8 X 10U~ .   Find
cm.2
the load (in pounds) necessary in order to produce an elongation of 0.5 -mm. in 1 meter.   The diameter of the wire is 1 millimeter.
4.   The modulus of shearing elasticity of a steel shaft is 11 X 106 pounds per square inch.   What force acting at the end of a lever 30 inches long will twist asunder the shaft if it is 0.5 inch in diameter?
6. A brass rod, 4 feet long and 1.5 inches in diameter, is twisted through an angle of 9 by a force of 1500 pounds acting 6 inches from the axis of the rod. If on removal of the stress the bar recovers its original position, calculate the modulus of shearing elasticity of the rod.
6.   Taking the data of the preceding problem find the force necessary to give an angle of twist of 2 to a rod 15 inches long, 0.5 inch in diameter.
7.   An elastic string of natural length I is stretched to twice its length when it supports a weight W.   The ends of the string are connected to two points at the same level and a distance d apart, while the weight W is attached to the middle of the string.   Find the position of equilibrium of the weight.
8.   A spider hangs from the ceiling by a thread which is stretched by the weight of the spider to twice its natural length.   Show that the work